February 14

Question 1

Show that 4 is not a primitive root modulo any prime p.

Question 2

Give a characterization of primes p for which the congruence
$x^4 + 1 \equiv 0 \mod p$
has at least one solution.

Question 3

How many solutions does the congruence
$1+x+x^2+\ldots+x^{23} \equiv 0 \mod 111$
have?

Question 4

Show that every prime divisor p>3 of
$n^2+n+1$
is congruent to 1 modulo 6.